Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

(1)

HAL Id: hal-01719592

https://hal.archives-ouvertes.fr/hal-01719592

Submitted on 28 Feb 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long

wave length asymptotics

Gabriele Barbagallo, Angela Madeo, Marco Valerio d’Agostino, Rafael Abreu, Ionel-Dumitrel Ghiba, Patrizio Neff

To cite this version:

Gabriele Barbagallo, Angela Madeo, Marco Valerio d’Agostino, Rafael Abreu, Ionel-Dumitrel Ghiba, et al.. Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics. International Journal of Solids and Structures, Elsevier, 2017, 120, pp.7-30. �10.1016/j.ijsolstr.2017.01.030�. �hal-01719592�

(2)

Transparent anisotropy for the relaxed micromorphic model:

macroscopic consistency conditions and long wave length asymptotics

Gabriele Barbagallo

¹

, Angela Madeo

²

, Marco Valerio d’Agostino

³

, Rafael Abreu

⁴

, Ionel-Dumitrel Ghiba

⁵

and Patrizio Neff

⁶

February 28, 2018

“Pluralitas non est ponenda sine necessitate.” - “Plurality should not to be supposed without necessity.”

John Duns Scotus - “Ordinatio”

Abstract

In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micromorphic model, also introducing their second order counterpart by using a Voigt-type vector notation.

In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit caseLc→0(which corresponds to considering very large specimens of a microstructured metamaterial the meso- and micro- coefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

Keywords: relaxed micromorphic model, anisotropy, arithmetic mean, geometric mean, harmonic mean, Reuss-bound, Voigt-bound, generalized continuum models, long wavelength limit, macroscopic consistency, Cauchy continuum, homogenization, multi-scale modeling, parameter identification, non-redundant model AMS 2010 subject classification: 74A10 (stress), 74A30 (nonsimple materials), 74A35 (polar materials), 74A60 (micromechanical theories), 74B05 (classical linear elasticity), 74E10 (anisotropy), 74E15 (crystalline structure), 74M25 (micromechanics), 74Q15 (effective constitutive equations)

1Gabriele Barbagallo, gabriele.barbagallo@insa-lyon.fr, LaMCoS-CNRS & LGCIE, INSA-Lyon, Universitité de Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne cedex, France

2Angela Madeo, corresponding author, angela.madeo@insa-lyon.fr, LGCIE, INSA-Lyon, Université de Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne cedex, France

3Marco Valerio d’Agostino, marco-valerio.dagostino@insa-lyon.fr, LGCIE, INSA-Lyon, Université de Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne cedex, France

4Rafael Abreu, abreu@uni-muenster.de, Institut für Geophysik, Westfälische Wilhelms-Universität Münster, Corrensstraße 24, 48149, Münster, Germany

5Ionel-Dumitrel Ghiba, dumitrel.ghiba@uni-due.de, dumitrel.ghiba@uaic.ro, Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany; Alexandru Ioan Cuza University of Iaşi, Department of Mathematics, Blvd. Carol I, no. 11, 700506 Iaşi, Romania; and Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi Branch, 700505 Iaşi.

6Patrizio Neff, patrizio.neff@uni-due.de, Head of Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Mathematik-Carrée, Thea-Leymann-Straße 9, 45127 Essen

(3)

1 Prelude

Modeling in continuum mechanics is an art encompassing mathematics, mechanics, physics and experiments.

Many researchers have been attracted to the field of generalized continuum mechanics, following the works of the masters Mindlin and Eringen and have dealt with the description of particular aspects of generalized continuum theories, usually introducing “ad hoc” terms to provide sensational additional effects. That has been done, while fundamental questions concerning the range of applicability or the descriptive power of generalized continuum mechanics had not been settled leading to an understandable disdain of the majority of researchers in continuum mechanics for these models. We, on the contrary, believe in the usefulness of generalized continuum mechanical models but at the same time we are aware of their current shortcomings.

We are deeply convinced that scientific advancements do not consist in producing a zoo of possibilities and to combine more effects (which are themselves not yet properly understood), but in reducing complexity and in explaining in simpler terms previously non-connected ideas without losing the accuracy of the mathematical description of the physical problem we are interested in.

A major guidance for enlightened modeling certainly comes from the experimental side. Basing ourselves on the phenomena we want to describe, we should not use superfluous information (superfluous because in practice, it cannot be determined) and, among valid competing hypotheses, the one with the simplest assumptions should be selected.

In this work we deal with the anisotropic relaxed micromorphic model in this spirit directed towards simplification. Whether we have achieved a step into this direction must be judged by our readers.

This paper originated from the need of setting up a transparent theory for the description of anisotropic materials with embedded microstructures.

Recent papers [43, 44] provided the evidence that the relaxed micromorphic model, even when restricted to the isotropic case, is usable to characterize the mechanical behavior of metamaterials with unorthodox dynamical properties. More precisely, it has been shown that the isotropic relaxed micromorphic model can be effectively used to model band-gap metamaterials, i.e. microstructured materials which are able to “stop”

the propagation of elastic waves due to local resonances at the level of the microstructure.

The enormous advantage of using the relaxed micromorphic model for the description of such metamaterials is undoubtedly that of mastering the behavior of complex media via the introduction of few elastic coefficients (Young modulus, Poisson ratio and few extra microstructure-related homogenized coefficients).

This simplified modeling of metamaterials allows to open the door towards the conception of “metastruc- tures”, i.e. structures which are made up of metamaterials as basic building blocks and which preserve their unconventional behavior at the scale of the structure (i.e. wave absorption).

The successful fitting of the isotropic relaxed micromorphic model on actual metamaterials was already been provided in [43,44]. Nevertheless, such studies have also suggested the need of generalizing the theoretical framework to the anisotropic case, to describe in detail the mechanical behavior of periodic and quasi-periodic metamaterials.

In this paper we provide such theoretical framework with the aim of readily applying it in a forthcom- ing paper which will be focused on the fitting of the anisotropic relaxed micromorphic model on actual metamaterials with low degree of anisotropy.

2 Introduction

Recent years have seen a colossal increase of interest in so called generalized or enriched continuum models.

This exponential growth is mainly due to the need felt to incorporate, viewed from the phenomenological level, additional features like the discreteness of matter, characteristic length scales, dispersion of waves, among others. All such features are not captured by standard elasticity approaches. The idea of using generalized continuum models to account for the homogenized behavior of microstructured materials has extensively been exploited in the last years (see e.g [21–25]). One of the most known generalized continuum models is the micromorphic continuum model introduced by Mindlin and Eringen [11, 16–19, 54] in the early sixties of the last century. It includes many special cases among which the much older Cosserat-type models [5, 12, 20, 34, 35, 40, 64, 70, 71].

In this paper we do not present the historic development of enriched continua, referring the reader to [52, 69] for this purpose. Furthermore, we restrict our attention to the linearized framework noting that the first

(5)

3D body

Ω

x _ϕ_(Ω)

Figure 1: Enriched kinematics for a micromorphic continuum. The macroscopic deformation of the body Ω ⊂ R³ is described by ϕ : Ω ⊂ R³ → R³. In each macroscopic material point x ∈ Ω there is a substructure attached. This substructure has the possibility to shear, stretch and shrink and is described by an affine mapping1+P. Decisive is the constitutive choice of the strain energy density which couples the macroscopic displacement u and the micro-distortion P. Our new relaxed micromorphic model introduces the weakest possible constitutive coupling still giving a well-posed model.

existence result for the geometrically nonlinear static case has been obtained in [34], which includes a previous result for the nonlinear Cosserat model [62]. For more details about existence results for micromorphic models at finite deformations, we refer the reader to [40, 63, 65, 66]. Further existence results are supplied in [14, 15, 50, 51]. There are many applications treated within the nonlinear micromorphic framework, among which we limit ourselves to mention [29–33, 36, 41, 53, 78–80, 86, 87].

In the micromorphic model, it is the kinematics which is enriched by introducing an additional field of non-symmetric micro-distortions P : Ω⊂R³ →R^3×3, beyond the classical macroscopic displacement u : Ω ⊂ R³ → R³ (see Fig. 1). Then, a non-symmetric elastic (relative) distortion e = ∇u − P can be defined and the modeling proceeds by obtaining the constitutive relations linking elastic-distortions to stresses and by postulating a balance equation for the micro-distortion field P. All such steps might be preferably done in a variational framework, involving the third order curvature tensor (the micro-distortion gradient) ∇P, so that only energy contributions need to be defined a priori. For the dynamic case, one adds in the Hamiltonian the so-called micro-inertia density contributions, acting on the time derivatives of micro-distortion terms P_,t.

In principle, the modeling framework for the micromorphic approach had been completed by Eringen, Mindlin, in the references already cited, and Germain [26]. Mindlin and Eringen also provided extensions of the micromorphic model to anisotropy even if such anisotropic models are almost impossible to be applied to real cases, due to the impressive number of coefficients provided (498 coefficients in the general anisotropic case).

The existence and uniqueness questions for the linear micromorphic model have been completely settled both for the static and dynamic case, based on the assumption of uniform positive definiteness of the appearing constitutive elasticity tensors. However, the over-reliance on uniform positive definiteness, we believe, has blinded the eyes for the real possibilities inherent in the micromorphic model. These possibilities have been consistently overlooked until very recently, when, in a series of articles [28, 46, 47, 68, 69], we have introduced the novel concept of relaxed micromorphic continuum. This model provides a drastic reduction of the number of constitutive coefficients with respect to Mindlin-Eringens’s one while remaining well-posed.

Unlike Mindlin-Eringen’s model, the relaxed model mainly works withsymmetric elastic (relative) strains ε_e:= sym (∇u− P), so that standard 4^th order symmetric elasticity tensors can be used in order to define elastic stresses. Moreover, regarding the curvature, the relaxed model considers thesecond order dislocation-density tensor α=−CurlP instead of the third order curvature tensor ∇P with the effect (among others) that the description of the anisotropy of curvature only needs 4^th order tensors, instead of 6^thorder ones.

A fundamental contribution of the relaxed micromorphic model is given by the fact that well-posedness results have been proven [69] also for the case where the strain energy density violates strict positive- definiteness⁷. In other words, even if the relaxed micromorphic model can be apparently seen as a particular

7It has to be noted that our new approach is only formally included in the standard Mindlin-Eringen micromorphic model since we consistently give up uniform positive-definiteness in the elastic distortioneand the curvature tensor∇P which are

(6)

case of the Mindlin-Eringen model by suitably setting some constitutive parameters of their model to zero (see [47, p. 555]), such choice is not acceptable in the Mindlin-Eringen setting due to the loss of positive- definiteness of the energy. Nevertheless, it is exactly this feature which makes the relaxed micromorphic model unique for the description of a wealth of unorthodox material behaviors. The existence results proposed in [69], as well as the drastic reduction of the number of the introduced elastic coefficients, allowed us to open the way to the application of the relaxed micromorphic model to cases of real interest.

Indeed, the relaxed micromorphic model has already been a source of inspiration for researchers working on granular materials [55].⁸ Moreover, the clear and transparent application of the relaxed micromorphic model in the isotropic case has recently been successfully achieved for the description of band-gap metamaterials (see [43, 44]).

As a matter of fact, the isotropic relaxed micromorphic model has proven its ability to fit the dispersion curves of phononic crystals for large windows of frequencies and wavelengths, arriving down to wavelengths which are comparable to the size of the unit cell. The most interesting aspect of the description of such metamaterials via the relaxed micromorphic model is undoubtedly that of predicting their macroscopic dynamical response through the introduction of few macroscopic elastic coefficients which are independent of the frequency.

This means that the coefficients of the relaxed micromorphic model can be seen astrue material parameters, exactly as it is the case for the Young modulus and the Poisson ratio when dealing with classical materials.

Of course, in order to extend the range of applicability of the relaxed model to a wider class of actual metamaterials, the model must be generalized to the anisotropic setting. This generalization is the principal aim of the present work.

In this paper, we want to present such an approach to anisotropy for the relaxed micromorphic model.

Our modeling perspective is to simplify as much as possible, and indeed to reduce to an essential minimum, the bewildering possibilities of the standard micromorphic model. Indeed, there is no point in exclaiming happily that the standard micromorphic model has more than 1000 constitutive coefficients which need to be determined. The true aim of modeling should consist of the opposite: discard all unclear complications without compromising the essence of the model. We believe that the relaxed micromorphic model is just going in this direction, thereby opening the way to transparent experimental campaigns for the determination of the remaining fewer extra parameters.

instead strictly requested in the standard model in order to have well-posedness. For example, controlling only the elastic strain εe= sym (∇u−P)in the energy does not locally control the elastic distortione= ∇u−P and working with CurlP does not control the curvature∇P.

8Although delighted by the fact of understanding that the relaxed micromorphic model might be of use for granular mechanics, we believe that some complements of information must be given in order to interpret the results of [55] in the clearest possible way. In [55] the authors use micro-macro upscaling techniques for granular assemblies arriving to a standard Mindlin-Eringen type model at the homogenized scale (see [55, p.224, eq. (43)]. The authors observe that: “remarkably, the nonzero components in Mindlin’s stiffness tensors are the same as the non-zero components derived from the present model”. Then, the authors present in equation (66) a constitutive choice of the microscopic parameters which goes in the sense of setting to zero the parameters of Mindlin’s model in order to get close to the relaxed micromorphic model. Such constitutive choice is not justified neither by telling that the scope is to recover the relaxed micromorphic model nor on clear microscopic-based arguments that would shed additional light on the understanding of microstructure-related effects.

Afterwards, the authors present [55, p.231, Fig. 5] two parametric studies on the parametersβmM andβsM. The parameter βmM is an analogous of the Cosserat couple modulusµcand is once again seen to be determinant for the onset of band gaps.

On the other hand, the parameterβsM is the one that, being non vanishing, still makes a difference between Mindlin-Eringen’s and our relaxed model. The authors then present a parametric study lettingβsM to zero, which indeed means that they are recovering the relaxed micromorphic model as a limit case. Nevertheless, except some mostly confusing sentences referring to our paper [47], such fundamental observation are not made at any point of the paper [55].

It should be clearly stated that, by means of the proposed parametric study, they are trying to approach the relaxed micromorphic model and that, although the corresponding choice of the parameters is not allowed in Mindlin-Eringen theory, the well-posedness is still guaranteed. Moreover, it should have been clearly stated that therelaxed micromorphic modelis the only generalized non-local continuum model, among those currently used, which is able to predictcomplete frequency band-gaps[28, 46, 47, 68].

Finally, and this would be for us the main advancement related to the paper [55], a clear microscopic-based interpretation of the fact of setting to zero the opportune parameters in Mindlin’s theory would be necessary in further works since it is not currently done in [55]. Of course, the fact of setting to zero some macroscopic parameters leads to some conditions on some microscopic parameters, but which is the physical interpretation of such conditions on micro-parameters?

In summary, the same goal of clarity that we try to pursue in this paper should be, from our point of view, shared by the highest possible number of researchers in order to proceed in the direction of a global advancement of knowledge.

(7)

The plan of the paper is as follows.

• We first recall the standard micromorphic model and contrast it with our new relaxed model. We also show that our relaxed micromorphic model supports a clear group-invariant framework, opening the way to the introduction of anisotropy classes.

• We present our favored description of anisotropy regarding the higher order contribution in CurlP. Thereby we split CurlP as sym CurlP + skew CurlP and let a classical fourth order tensor act only on sym CurlP∈Sym(3)together with another tensor with only 6 parameters acting on skew CurlP ∈ so(3).

• We consider the long-wavelength limit (characteristic lengthL_c→0) which must coincide with a linear elastic model that has lost any characteristic length (sometimes called internal variable model). From this hypothesis, we are able to relate coefficients of the micromorphic scale to the macroscopic ones.

The result is a convincing homogenization formula for all considered anisotropy classes. This is also done using classical Voigt-notation in order to facilitate future applications. As already stated, this homogenization formula relating micro and macro parameters is one of the main results of the present work, since it opens the way to the application of the model to actual metamaterials via the realization of standard mechanical tests on “large” specimens.

• We study the format of a possible anisotropic local rotational coupling term acting on skew (∇u −P).

In this respect we also investigate some possibilities of approximating an anisotropic coupling by an isotropic one.

• We consider the formal limitL_c→ ∞and show that it corresponds to a “zoom” into the micro-structure.

Our relaxed model supports also a clear interpretation for that regime.

• We end our paper by showing that the standard Mindlin-Eringen micromorphic model does not support the clear relation between macroscopic and microscopic elasticity moduli which is instead provided by our simplified anisotropic relaxed model.

3 Notational agreement

Throughout this paper Latin subscripts take the values 1,2,3 while Greek subscripts take the values 1,2,3,4,5,6and we adopt the Einstein convention of sum over repeated indices if not differently specified.

We denote byR^3×3 the set of real 3×3 second order tensors and by R^3×3×3 the set of real 3×3×3 third order tensors. The standard Euclidean scalar product on R^3×3 is given by

X, Y

R^3×3 = tr(X·Y^T) and, thus, the Frobenius tensor norm iskXk²=

X, X

R^3×3. Moreover, the identity tensor onR^3×3will be denoted by1, so that tr(X) =

X,1

. We adopt the usual abbreviations of Lie-algebra theory, i.e.:

• Sym(3) :={X ∈R^3×3|X^T =X} denotes the vector-space of all symmetric3×3matrices

• so(3) :={X ∈R^3×3|X^T =−X} is the Lie-algebra of skew symmetric tensors

• sl(3) :={X ∈R^3×3|tr(X) = 0}is the Lie-algebra of traceless tensors

• R^3×3'gl(3) ={sl(3)∩Sym(3)} ⊕so(3)⊕R·1is theorthogonal Cartan-decomposition of the Lie-algebra For allX ∈R^3×3, we consider the decomposition

X = dev symX+ skewX+1

3tr(X)1 (1)

where:

• symX =¹₂(X^T +X)∈Sym(3)is the symmetric part,

• skewX= ¹₂(X−X^T)∈so(3)is the skew-symmetric part,

(8)

• devX=X−¹₃tr(X)1∈sl(3)is the deviatoric part . Throughout all the paper, we denote:

• the sixth order tensorsLb:R^3×3×3→R^3×3×3 by a hat

• the fourth order tensorsC:R^3×3→R^3×3 by overline

• without superscripts, i.e.C, the classical fourth order tensors acting only on symmetric matrices C: Sym(3)→Sym(3)or skew-symmetric onesCc:so(3)→so(3)

• the second order tensorsCe:R⁶→R⁶ orCe:R³→R³ appearing as elastic stiffness by a tilde.

We denote byCX the linear application of a4^thorder tensor to a2^ndorder tensor and also for the linear application of a6^th order tensorLb to a3^rd order tensor. In symbols:

CX

ij =CijhkXhk, LbA

ijh=LbijhpqrApqr. (2)

The operation of simple contraction between tensors of suitable order is denoted by a central dot, for example:

Ce·v

i

=Ceijv_j, Ce·X

ij

=CeihX_hj. (3)

Typical conventions for differential operations are implied, such as a comma followed by a subscript to denote the partial derivative with respect to the corresponding Cartesian coordinate, i. e. (·)_,j = ^∂(·)_∂x

j. Given a skew-symmetric matrixA∈so(3)we consider:

A=





0 A₁₂ A₁₃

−A12 0 A23

−A13 −A23 0



, axl A

= (−A23, A13,−A12)^T. (4)

ore equivalently in index notation:

axl A

k=−1

2ijkAij= 1

2kijAji, (5)

where is the Levi-Civita third order permutation tensor.

4 A review on the micromorphic approach

In this section we recall the general anisotropic setting of classical Mindlin-Eringen micromorphic elasticity, as well as that of relaxed micromorphic elasticity. We show that, given its intrinsic formulation, the relaxed micromorphic model features 93 coefficients instead of Mindlin/Eringen 498.

In subsection 4.2.1, a further reduction of coefficient is proposed for those cases in which one wants to feature a symmetric stress.

In subsection 4.2.2, it is shown that the most general form of the relaxed curvature energy (in the anisotropic setting) which satisfies certain additional invariance requirements features 21+7=28 coefficients instead of the 378 featured by the classical Mindlin-Eringen model. Moreover some further simplification of the curvature energy are proposed, up to arriving to the isotropic case in which the curvature energy only shows 3 coefficients. As a matter of fact, we propose a consistent framework for the definition of the curvature energy of the relaxed micromorphic model which is fully consistent with invariance arguments.

Such clear theoretical framework is of primordial importance for the introduction of suitable constitutive expressions for the curvature energy. Nevertheless, it is likely that, in a first instance, non-local effects in real metamaterials can be controlled via the introduction of very few characteristic lengths. For this reason, the maximum generality of the anisotropic setting for the curvature could find effective applications only in a second instance, when the most important step of the identification of the elastic coefficients Ce, Cmicro

andCc will be achieved on a suitable class of targeted metamaterials.

(9)

In subsection 4.2.3, the general anisotropic setting for the kinetic energy to be used in the relaxed micromorphic model is provided. This step is strongly complementary to the constitutive choice for the static case featured by equation (8). Indeed, if some deformation mechanism are introduced in the definition of the strain energy densities, analogous inertiae must be introduced in the kinetic energy to have a well-posed problem in the dynamical case. This step is essential to securely proceed towards controllable applications on actual metamaterials subjected to dynamical loading.

4.1 The standard Mindlin-Eringen model

The elastic energy of the general anisotropic centro-symmetric micromorphic model in the sense of Mindlin- Eringen (see [54] and [17, p. 270, eq. 7.1.4]) can be represented as:

W =1 2

Ce(∇u− P),(∇u −P)

R^3×3

| {z }

full anisotropic elastic−energy +1

2

Cmicro symP,symP

R^3×3

| {z }

micro−self−energy

(6)

+1 2

Ecross (∇u− P),symP

R^3×3

| {z }

anisotropic cross−coupling

+µL²_c 2

Lbaniso∇P,∇P

R^3×3×3

| {z }

full anisotropic curvature ,

whereCe:R^3×3→R^3×3is a4^thorder micromorphic elasticity tensor which has at most 45 independent coefficients and which acts on thenon-symmetric elastic distortione= ∇u−P andEcross :R^3×3→Sym(3) is a 4^th order cross-coupling tensor with the symmetry Ecross

ijkl = Ecross

jikl having at most 54 independent coefficients. The fourth order tensor Cmicro : Sym(3) →Sym(3) has the classical 21 independent coefficients of classical elasticity, whileLbaniso:R^3×3×3→R^3×3×3is a6^thorder tensor that shows an astonish- ing 378 parameters. The parameterµ >0is a typical shear modulus andLc>0is one characteristic length, while Lbaniso is, accordingly, dimensionless. Here, for simplicity, we have assumed just a decoupled format of the energy: mixed terms of strain and curvature have been discarded by assuming centro-symmetry.

Counting the number of coefficients we have 45 + 21 + 54 + 378 = 498independent coefficients.

If we assume an isotropic behavior of the curvature we obtain:

W =1 2

C^e(∇u− P),(∇u −P)

R^3×3

| {z }

full anisotropic elastic−energy +1

2

C^micro symP,symP

R^3×3

| {z }

(7)

+1 2

Ecross (∇u− P),symP

R^3×3

| {z }

anisotropic cross−coupling

+µL²_c 2

Lbiso∇P,∇P

R^3×3×3

| {z }

isotropic curvature ,

where the6^thorder tensorLbisohas still 11 independent non-dimensional constants [17]. This can be explained considering that the general isotropic6^thorder tensor has 15 coefficients which, considering that in a quadratic form representation we can assume a major symmetry of the type Lbijklmn = Lblmnijk, reduce to 11 (see [57, 83]).⁹ On the other hand, the local energy has 7 independent coefficients in the isotropic case: Ce has 3, Cmicro ∼2, Ecross ∼2 adding up to the usual 18 constitutive coefficients to be determined in the isotropic case.

One of the major obstacles in using the micromorphic approach for specific materials is the impossibility to determine such multitude of new material coefficients. Not only is the huge number a technical problem, but also the interpretation of coefficients is problematic [8–10]. Some of these coefficients are size-dependent while others are not. A purely formal approach, as is often done, cannot be the final answer.

4.2 The relaxed micromorphic model

Our novel relaxed micromorphic model endows Mindlin-Eringen’s representation with more geometric structure. SinceEcross is difficult to interpret, it is discarded right-away. Nevertheless, the structure of the model

9The 11 coefficients of the curvature in the isotropic case reduce to 5 in the particular case of second gradient elasticity (see [13]) which is obtained from a micromorphic model by setting P =∇u.

(10)

continues to be very rich. We write:

W =1 2

Cesym (∇u−P), sym (∇u− P)

R^3×3

| {z }

anisotropic elastic−energy

+1 2

Cmicro symP,symP

R^3×3

| {z }

(8)

+1 2

Cc skew (∇u− P),skew (∇u− P)

R^3×3

| {z }

invariant local anisotropic rotational elastic coupling

+µL²_c 2

Laniso CurlP,CurlP

R^3×3

| {z }

curvature

.

The second order tensor α := −CurlP is usually called the dislocation density tensor.¹⁰ Here Ce, Cmicro : Sym(3) → Sym(3)are both classical4^thorder elasticity tensorsacting on symmetric second order tensorsonly: Ce acts on thesymmetric elastic strainε_e:= sym (∇u− P)andCmicro acts on thesymmetric micro-strain symP and both map to symmetric tensors. The tensorCc:so(3)→so(3) is a4^thorder tensor that acts only on skew-symmetric matrices and yields only skew-symmetric tensors and Laniso:R^3×3→R^3×3is a dimensionless4^thorder tensor with at most 45 constants. Counting coefficients we now have 21+21+6+45=93, instead of Mindlin-Eringen’s 498 coefficients. The main advantage at this stage is that our Ce, unlike Ce, possesses all the symmetries that are peculiar of the classical elasticity tensors acting on sym∇u.

The large number of isotropic constants in the standard Mindlin-Eringen model has always been of concern. Previous attempts to endow the Mindlin-Eringen model with more structure include Koh’s [37,77] so- called micro-isotropy postulate which requires, among others, that symσis an isotropic function of sym∇u only. This reduces the number of isotropic coefficient also to 5 (similarly to our relaxed model) but the fact of connecting symσto sym∇u only cannot be considered to be a well-grounded hypothesis.

Considering the energy in equation (8), the resulting elastic stress is:

σ(∇u , P) = Cesym (∇u −P) +Cc skew (∇u −P), (9) which is solely related to elastic distortions e= ∇u −P. One of the main results of the present paper is to provide a simple but effective homogenization formula which relates the elastic tensorsCeandCmicro to the macroscopic elastic properties of the considered medium that will be encoded in the effective elastic tensor C^macro.

The derivation of the macroscopic consistency condition we propose in the present paper is of primary importance for an effective application of the proposed model to cases of real interest.

Indeed, the basic idea is that of considering a sample of a specific microstructured material which is large enough to let the effect of the underlying microstructure being negligible. On this large sample, standard mechanical tests can be performed to allow for the unique determination of the elastic coefficients Cmacro.

The existence of our formula relating Cmacro (which is well known) to Ce and Cmicro (which are still unknown), allows to further reduce the number of coefficients that need to be determined to unequivocally characterize the mechanical behavior of microstructured materials.

This unique feature of our relaxed model gives again more credibility to the relaxed approach by opening the way to a clear experimental campaign to determine some of the new micromorphic elastic constants.

In the general anisotropic micromorphic model initially proposed by Mindlin-Eringen [19] the question of parameter identification has already been treated. However, the resulting interpretation of the material constants, as well as their connection to the classical anisotropy formulation of linear elasticity, is still not settled satisfactorily, and presumably impossible.

As already seen, in our relaxed model the complexity of the general micromorphic model has been deci- sively reduced, featuring basically only symmetric strain-like variables and the Curl of the micro-distortion P. However, the relaxed model is still general enough to include the full micro-stretch as well as the full Cosserat micro-polar model, see [69]. Furthermore, well-posedness results for the static and dynamic cases have been provided in [69] making decisive use of recently established coercive inequalities, generalizing Korn’s inequality to incompatible tensor fields [2, 61, 74–76].

10The dislocation tensor is defined asαij=−( CurlP)_ij=−Pih,kjhk, whereis the Levi-Civita tensor.

(11)

Furthermore, certain limiting cases of the anisotropic relaxed micromorphic model give as a result other micromorphic models, e.g. the Cosserat model, the micro-dilation theory, the micro-incompressible micromorphic model, the micro-stretch theory and the microstrain model) as it is shown in Appendix A.1. Instead, the second gradient model cannot be found as a limiting case differently from what happens in the Eringen Mindlin micromorphic model, see Appendix A.2 for the one dimensional case.

4.2.1 Possible symmetry of the relaxed micromorphic stress

In this subsection, we recall some arguments that allow the possibility of featuring a symmetric stress tensor for the relaxed micromorphic model by setting the 6 components of the tensorCcto be vanishing. Considering the scalar product

X, Y

= tr(X·Y^T), we start by noticing that, given the definition of the fourth order tensors Ce and Cc, they respect a generalized version of the orthogonal decompositionof second order tensors (X = symX⊕skewX), in the sense that:

sym [CesymX+Cc skewX] =CesymX, (10) skew [CesymX+Cc skewX] =Cc skewX .

We recall that the elastic stress of the relaxed micromorphic model is:

σ(∇u , P) = Cesym (∇u −P) +Cc skew (∇u −P), (11) so that skew-symmetry of the elastic stressσis entirely controlled by the rotational coupling tensorCc since, relying on formulas (10), we have

skewσ= skew [Ce sym (∇u− P) +Cc skew (∇u− P)] =Cc skew (∇u −P). (12) For a positive definite coupling tensorCc, we note that skew-symmetric stressesskewσ6= 0occur if and only if skew (∇u− P)6= 0.

IfCc≡0, the elastic Cauchy stressσsatisfiesBoltzmann’s axiom of symmetry of force stresses.

In addition, for Cc ≡0, the elastic distortione = ∇u − P can be non-symmetric, while the elastic stressσremains symmetric.¹¹

In [78] the authors have introduced the original and important notion of non-redundant strain mea- suresin the micromorphic continuum. As it turns out, the relaxed micromorphic model with zero rotational coupling tensor Cc ≡ 0 is a non-redundant micromorphic formulation. Conversely, the standard Mindlin- Eringen model remainsredundant, as does the linear Cosserat model.

With Boltzmann’s axiom, which is in sharp contrast to standard micromorphic models, the model would feature symmetric force-stress tensors. Such an assumption has been made, for example, by Teisseyre [84, 85]

in his model for the description of seismic wave propagation phenomena (for the use of micromorphic models for earthquake modeling see also the discussion in [60]).

It must also be observed that the relaxed micromorphic model can be used withCc positive semi-definite or indeed zero (in the isotropic caseµc= 0), while we always assume thatCe,Cmicro (and laterCmacro) are strictly positive definite tensors. Assuming that Ce andCmicroare positive definite tensors means that:

∃c⁺_e >0 : ∀S∈Sym(3) :

CeS, S

R^3×3 ≥c⁺_ekSk²

R^3×3. (13)

11Therefore, usingCc= 0is similar to the Reuss-bound approach in homogenization theory in which the guiding assumption is that the stress fields are taken constant but fluctuations in strain are allowed. Here, analogously, we would assume symmetric stressesσbut non-symmetric distortion-fluctuations ine= ∇u −P. Voigt (see [89, p.596]) already discussed non-symmetric states of distortion. However, we can supply some further support for usingCc≡0. Indeed as Kröner notes [38],“asymmetric stress tensors only come under consideration when a distribution of rotational moments acts upon the body externally, which is excluded here. The question of whether the (...) rotations produces stresses can also be answered. We must first exclude asymmetric stress tensors, since they contradict the laws of equilibrium in the theory of elasticity”. Furthermore, Kunin [39, p.

21] states the following theorem: in the nonlocal theory of a linear elastic medium of simple structure with finite action-at-a- distance, it is always possible to introduce a symmetric stress tensor and an energy density, which can be expressed in terms of stress and strain in the usual way.

(12)

In sharp contrast to the standard Mindlin-Eringen format, we assume for the rotational coupling tensor Cc

only positive semi-definiteness, i.e:

∀A∈so(3) :

CcA, A

R^3×3 ≥0. (14)

As already noted, this allows the rotational coupling tensor Cc to vanish, in which case the relaxed micromorphic model isnon-redundant[78].¹²

The reader might ask himself: how is it possible that the rotational coupling tensorCc can be absent but the resulting model is still well-posed? This is possible because in that case, the skew-symmetric part of P is not controlled locally but as a result of the boundary value problem and boundary conditions. In this sense, allowing for Cc ≡0 is one of the decisive new possibilities offered by the relaxed micromorphic model.

However, in [47] it has been shown that in the isotropic case (Cc =µ_c1) the presence of Cc allows to control the onset of band-gaps. In section 5 we discuss the possible forms thatCc may have for certain given anisotropy classes.

4.2.2 Microscopic curvature

In the general micromorphic model, the curvature energy term is of the form:

Wcurv=Wcurv(∇P), (15)

In our relaxed framework, we assume that it depends only on the second order dislocation density tensor through:

W_curv^relax=W_curv^relax( CurlP). (16)

First, we remark here that the relaxed micromorphic curvature expression can also be written as:

CurlP =−Curl (∇u− P), (17)

because CurlP is invariant under P → P+∇ϑ, see [68].

Now, we need to shortly discuss that such a reduced formulation is fully able to be treated in an invariant setting. To this end, let P: Ω ⊂R³ →R^3×3 be the micro-distortion field to which we apply the following coordinate transformation (generating the so-called Rayleigh-action on it [1]):

P^#(ξ) :=Q·

P(Q^T ·ξ)

·Q^T, x=Q^T·ξ, (18) for given Q ∈ SO(3). Transforming the displacement to a rotated reference and spatial configuration, we have:

u^#(ξ) :=Q·u(Q^T·ξ), ∇_ξu^#(ξ) =Q· ∇_xu(Q^T·ξ)·Q^T, (19) thus, we require that P transforms as ∇u under simultaneous rotations of the reference and spatial config- urations. Then it can be shown [58] that:

Curl_ξP^#(ξ) =Q·[ Curl_xP(Q^T ·ξ)]·Q^T. (20) For the description of anisotropy in the curvature energy we require form-invarianceof expression (16) under the transformation (18) with respect to all rotations Q ∈ G-material symmetry group. Taking (20) into account, this means

∀Q∈ G −material symmetry group: W_curv^relax(Q^T · CurlP·Q) =W_curv^relax( CurlP). (21) In the same spirit as done with the local energy terms, a first simplification of the curvature expression, which is consistent with the invariance condition (21) is given by:

W_curv^relax( CurlP) =µL²_c 2

h

Lesym CurlP,sym CurlP +

Lc skew CurlP, skew CurlPi

. (22)

12In Misra et al. [55] the rotational couplingCcis related to the tangential stiffness between grains. This is consistent with Shimbo’s law [81] relating the rotational stiffness to the internal friction. We need to remark that friction is, strictly speaking, a dissipative effect outside purely elastic response.

(13)

Here, Le : Sym(3)→ Sym(3) is a classical, positive definite elasticity tensor with at most 21 independent (non-dimensional) coefficients andLc :so(3)→so(3)is a positive definite tensor with at most 6 independent (non-dimensional) coefficients. Taking isotropy into account, the total number of coefficients reduces to 3, while in the cubic case we have 4 coefficients.

We can think of another reduction of the curvature expression which is fully consistent with group- invariance requirements. Let:

W_curv^relax=Wc_curv^relax( sym CurlP). (23) Considering the same transformation law (18) as before, the complete representation of anisotropy in terms of representation (23) is easy. Indeed, we may employ the classical format of the 4^thorder elasticity tensors to write:

Wc_curv^relax( sym CurlP) =µL²_c 2

Lesym CurlP,sym CurlP

. (24)

Here Le : Sym(3) → Sym(3) is a classical, positive definite elasticity tensor with at most 21 independent (non-dimensional) coefficients. The expression in (24) is certainly preferable for its simplicity for the treat- ment of anisotropic curvatures. However, it is not clear whether a formulation based on (24) can lead to mathematically well-posed results due to the current lack of a suitable coercive inequality for that case [2, 3].

Our guess at the moment is that it should work for the micro-incompressible case, in which the constraint trP = 0is appended. This case is reminiscent of gradient plasticity with plastic spin [14, 15, 74] in which the micro-distortion P is identified with the plastic distortion.

As explained in detail in [58], isotropy of the curvature energy is tantamount to requiringform-invariance of expression (16) under the transformation (18), i.e.:

W_curv^relax CurlξP^#(ξ)

=W_curv^relax( CurlxP(x)). (25)

Taking (20) into account, isotropy of the curvature is satisfied if and only if:

∀Q∈SO(3) : W_curv^relax Q·( CurlxP(x))·Q^T

=W_curv^relax( CurlxP(x)). (26) i.e. W_curv^relax must be an isotropic scalar function. We need to highlight the fact that CurlP is not just an arbitrary combination of first derivatives of P (and as such included in the standard Mindlin-Eringen most general anisotropic micromorphic format), but that the formulation in CurlP supports a completely invariant setting, as seen in [58], [64]. Since CurlP is a second order tensor, it allows us to discard the 6^th order tensors of classical Mindlin-Eringen micromorphic elasticity and to work instead with4^thorder tensors whose anisotropy classification is much easier and well-known [7].

In general, if we consider an isotropic curvature term, we obtain the following representation:

µL²_c 2

Liso CurlP,CurlP

R^3×3 =µL²_c 2

α₁kdev sym CurlPk²+α₂kskew CurlPk²+α₃[tr( CurlP)]² , (27) with scalar weighting parameters α1, α2, α3 ≥0. Since, in this paper, the curvature energy does not play a major role we will mostly just usekCurlPk², corresponding toα1, α2= 1 andα3= ¹₃. .

4.2.3 Micro-inertia density

The dynamical formulation of the proposed relaxed micromorphic model is obtained in the following way.

We define a joint Hamiltonian and obtain the equations from the postulate of stationary action. In order to generalize the kinetic energy density to the anisotropic micromorphic framework, we need to introduce a micro-inertia density contribution of the type:

1 2

JP_,t, P_,t

. (28)

Here J : R^3×3 → R^3×3 is the 4^th order micro-inertia density tensor with, in general, 45 independent coefficients. Eringen has added a conservation law for the micro-inertia density tensor J, but in this work we

(14)

assume a constant micro-inertia density tensorJas well as a constant mass densityρ. We assume throughout this paper that Jis positive definite, i.e.:

∃c⁺>0 : ∀X ∈:R^3×3:

JX, X

R^3×3 ≥c⁺kXk²_R3×3. (29) Considering dimensional consistency, we can always write the micro-inertia density tensorJas:

J=ρLb²_cJ0, (30) where J0:R^3×3→R^3×3 is dimensionless. Here, ρ >0 is the mean mass density[ρ] =kg/m³ andLbc ≥0 is another characteristic length[Lb_c] =m. We also propose a split of this micro-inertia density, similar to that adopted for the other elastic tensors like:

1 2

JP_,t, P_,t

= 1 2

JesymP_,t,symP_,t +1

2

Jc skewP_,t,skewP_,t

. (31)

Here, Je: Sym(3)→Sym(3)maps symmetric tensors into symmetric tensors whileJc :so(3)→so(3) maps skew-symmetric tensors to skew-symmetric tensors. We assume then that bothJeandJcare positive definite.

In the isotropic case, the micro-inertia density tensorJ0 can be represented by three dimensionless pa- rametersη₁, η₂, η₃>0 such that:

1 2

JP_,t, P_,t

= ρbL²_c 2

η₁kdevsymP_,tk²+η₂kskewP_,tk²+η₃(tr(P_,t))²

. (32)

4.2.4 Linear elasticity as upper energetic limit for the relaxed micromorphic model - statics The relaxed micromorphic model admits linear elasticity as an upper energetic limit for any characteristic length scaleLc>0. This can be seen by noticing that an admissible field for the micro-distortion P is always P = ∇u. Then, a standard minimization argument shows¹³:

min

(u, P)

Z

Ω

1 2

Ce sym (∇u −P),sym (∇u− P)

R^3×3+1 2

Cmicro symP,symP

R^3×3 (33) +1

2

Cc skew (∇u −P),skew (∇u −P)

R^3×3+µL²_c 2

Laniso CurlP,CurlP

R^3×3dx

≤ Z

Ω

1 2

Cmicro sym∇u ,sym∇u

R^3×3dx .

Thus, we see that the relaxed model is always energetically weaker than a linear elastic comparison material with elastic stiffnessC^micro for any given stiffnessC^e. This, again, is in contrast to the standard Mindlin-Eringen format which will, in general, generate arbitrary stiffer response as Lc → ∞and Ce→ ∞ simultaneously.

13The strict equality in (33) is trivial considering that replacing P=∇u on the left hand side and recalling thatCurl∇ϑ= 0.

On the other hand, the inequality can be justified thinking that a solution (u^∗, P^∗) of the relaxed micromorphic problem is a minimizer, in the sense thatW(u^∗, P^∗)≤W(u, P)for any admissible field(u, P). Hence, taking a generic field P = ∇u (which is of course admissible) justifies the equation (33).

(15)

4.3 Energy formulations and equilibrium equations for various symmetries

Gathering our findings, we propose the following representation of the energy for the relaxed anisotropic centro-symmetric model, which has maximally 21+21+6+21+6=75 independent coefficients:

W =1 2

Cesym (∇u−P), sym (∇u− P)

R^3×3

| {z }

anisotropic elastic−energy

+1 2

Cmicro symP,symP

R^3×3

| {z }

(34)

+1 2

Cc skew (∇u− P),skew (∇u− P)

R^3×3

| {z }

invariant local anisotropic rotational elastic coupling +µL²_c

2 L^e sym CurlP,sym CurlP

R^3×3+

L^c skew CurlP, skew CurlP

R^3×3

| {z }

curvature

.

This constitutive expression of the strain energy density for the relaxed micromorphic model is the most general one that can be provided in the anisotropic and centrosymmetric framework and already it provides a drastic reduction of the constitutive coefficients with respect to the standard Mindlin-Eringen model (75 coefficients against the 498 of Mindlin-Eringen). With a look towards immediate applications, it can be considered that non-local effects can be considered, in a first instance, to be isotropic, so that the curvature coefficients reduce from 21+6=27, to at most 2. We end up with a fully anisotropic model which features at most 51 parameters for describing:

• the full anisotropy at the microstructural level.

• the full anisotropy at the macroscopic level.

• the possibility of describing non-localities through the introduction of suitable characteristic lengths.

Of course, given particular metamaterials with particular symmetries, this number of parameters can be further reduced.

For example, the fully isotropic case requires to determine (Ce∼2, Cmicro ∼2,Cc ∼1,Le∼2,Lc ∼1) altogether 8 constitutive coefficients of which the rotational coupling coefficientµccan be set to zero to enforce symmetricelastic stressesσ. As seen before, Eringen’s formulation has 18 coefficients and Koh’s [37] micro- isotropic model has still 10.¹⁴ This simplified framework allowing to describe the full micro-macro anisotropy and the presence of non-localities via the introduction of “only” 51 parameters is of fundamental importance to proceed towards an enlightened characterization of the actual metamaterial.

Considering

ρLb²_c 2

J0 P_,t, P_,t

, (35)

as the micro-inertia term, the dynamical equilibrium equations for the anisotropic relaxed micromorphic model take the compact format:

ρ u_,tt= Div [Cesym (∇u −P) +Cc skew (∇u− P)],

ρLb²_cJ0P,tt=C^e sym (∇u −P) +C^c skew (∇u− P)−C^micro symP (36)

−µL²_cCurl (Lesym CurlP+Lc skew CurlP).

14Note that establishing positive-definiteness of the energy is now an easy matter as compared to [82]: we only need to require positive definiteness of the occurring standard 4^thorder tensorsCe,Cmicro,Cc,Le,Lc.

(16)

If we consider the isotropic case and the simplest curvature form, it is possible to reduce the relaxed representation to (see [43, 45–47, 69, 72]):

W =µeksym (∇u− P)k²+λe

2 tr ( sym (∇u −P))²

| {z }

isotropic elastic−energy

+µmicroksymPk²+λmicro

2 (tr ( symP))²

| {z }

(37)

+µckskew (∇u −P)k²

| {z } invariant local isotropic rotational elastic coupling

+ µL²_c

2 kCurlPk²

| {z } isotropic curvature

,

and the isotropic format of the micro-inertia becomes:

1 2

JP_,t, P_,t

= ρbL²_c 2

η₁kdevsymP_,tk²+η₂kskewP_,tk²+η₃(tr(P_,t))²

. (38)

Hence, the dynamical equilibrium equations for the isotropic relaxed micromorphic model take the form:

ρ u_,tt= Div [Cesym (∇u− P) +Cc skew (∇u−P)], (39) η1ρLb²_cdev sym [P,tt] = dev sym

Cesym (∇u−P)−Cmicro symP−µL²_cCurl CurlP ,

η₂ρLb²_c skew [P_,tt] =Cc skew (∇u− P)−µL²_cskew Curl CurlP ,

η3ρLb²_ctr [P,tt] = tr

Cesym (∇u −P)−Cmicro symP−µL²_cCurl CurlP . For more information about the dynamics of the relaxed micromorphic model see [48, 72].

4.4 Some considerations on the macroscopic consistency condition in the isotropic case

In this section, we want to recall some results concerning themacroscopic consistency conditionfor the relaxed micromorphic model in the isotropic case [62, 67].

Although such condition has already been derived in [62, 67] and even if it is only valid for the particular case of isotropy, we want to underline the idea which is behind such condition. Indeed, it is of fundamental importance to catch the power that the introduced homogenization formulas may have for an effective application of the relaxed micromorphic model. In section 6.2, we will present a generalization of such homogenization formulas to the fully anisotropic framework so opening the way for the effective mechanical characterization of a huge class of mechanical metamaterials.

The main idea, which is behind the determination of our homogenized formulas, is that of considering a very large sample of a given microstructured material. This sample must be large enough that the effect of the microstructure on the macroscopic behavior of the sample can be considered to be negligible.

Under this hypothesis, we can introduce a macroscopic elasticity tensorCmacro: Sym(3)→Sym(3)which best fits the macroscopic behavior of the sample and we can suppose that the material behavior can be described by classical linear elasticity with energy:

W =1 2

Cmacrosym∇u ,sym∇u

. (40)

The corresponding classical symmetric Cauchy stress is clearly defined as:

σ( sym∇u) =Cmacro sym∇u . (41)

For very large sample sizes, however, a scaling argument shows easily that therelativecharacteristic length scale L_c of the micromorphic model must vanish. Therefore, we have a way of comparing the classical

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

HAL Id: hal-01719592

https://hal.archives-ouvertes.fr/hal-01719592

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long

wave length asymptotics

Gabriele Barbagallo, Angela Madeo, Marco Valerio d’Agostino, Rafael Abreu, Ionel-Dumitrel Ghiba, Patrizio Neff

To cite this version:

Transparent anisotropy for the relaxed micromorphic model:

macroscopic consistency conditions and long wave length asymptotics

Gabriele Barbagallo

, Angela Madeo

, Marco Valerio d’Agostino

, Rafael Abreu

, Ionel-Dumitrel Ghiba

and Patrizio Neff

February 28, 2018

Contents

1 Prelude

2 Introduction

Ω

3 Notational agreement

4 A review on the micromorphic approach

4.1 The standard Mindlin-Eringen model

4.2 The relaxed micromorphic model

4.3 Energy formulations and equilibrium equations for various symmetries

4.4 Some considerations on the macroscopic consistency condition in the isotropic case